\documentclass{article}

\setlength{\topmargin}{0pt}
\setlength{\oddsidemargin}{0pt}
\setlength{\evensidemargin}{0pt}
\setlength{\textwidth}{6in}
\setlength{\voffset}{0pt}
\setlength{\headheight}{0pt}
\setlength{\headsep}{0pt}
\setlength{\textheight}{9in}

\usepackage{amsmath}
\usepackage{latexsym}
\usepackage{amssymb}

\newcommand{\ve}{\vec}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\lb}{\left[}
\newcommand{\rb}{\right]}
\newcommand{\lc}{\left\{}
\newcommand{\rc}{\right\}}
\DeclareMathOperator{\curl}{curl}
\DeclareMathOperator{\diver}{div}
\newcommand{\ra}{\rightarrow}
\renewcommand{\Re}{\mathbb{R}}
\newcommand{\Nat}{\mathbb{N}}
\newcommand{\sB}{\mathcal{B}}
\newcommand{\sC}{\mathcal{C}}
\newcommand{\sF}{\mathcal{F}}
\renewcommand{\phi}{\varphi}
\DeclareMathOperator{\Real}{Re}
\DeclareMathOperator{\Imag}{Im}

\DeclareMathOperator{\spn}{span}


\begin{document}

\begin{center}
\LARGE
Homework \#4.5  PRACTICE --- not collected or graded
\normalsize

\bigskip
taken from Math 205, Spring 2014 for Spring 2016 practice

\bigskip
%{\bf Due Friday, Oct.\ 12 in class}
\end{center}
You must show your work in order to receive full credit; correct answers with 
no, or insufficient work, may not receive credit.  Problems 1--4 cover
material of Sections 4.2--4.4; problems 5 \& 6 covered week 6 material,
sections 4.5 \& 4.6 and are omitted.

\bigskip

\normalsize

\begin{enumerate}

\bigskip
%{\bf WEEK 5 \rm }

\bigskip

\item Determine whether the following subsets of $C^2(\Re)$ are vector spaces 
or not (that is, whether they're subspaces of $C^2(\Re)$ or not).
\begin{enumerate}
\item The set of functions $f$ satisfying $f(1)=0$.
\item The set of functions $f$ satisfying $f(1)=-1$.
\end{enumerate}

\bigskip

\item Consider the matrix
\[
A= \left[\begin{array}{cccc}
-2 & 1 & 1 & -1  \\
1 & -1 & 1 & 2  \\
-1 & 0 & 2 & 1  
\end{array}\right] .
\]

\begin{enumerate}
\item Find the nullspace of $A$.
\item Now finding a spanning set for the nullspace of $A$.
\end{enumerate}


\bigskip


\item Consider the following set of three vectors in $\Re^3$
\[
\lc \lp 2,2,7 \rp, \lp 1,5,11 \rp, \lp -1,3,4 \rp \rc .
\]
Determine whether or not these vectors span $\Re^3$.  If
not, write one of the vectors as a linear combination
of the other two.

\bigskip

\item Consider the vectors $v_1 = \lp 6,1,-3 \rp$ and $v_2= \lp -3,3,2 \rp$ in $\Re^3$. Is $(3,4,-3)$ in $\spn\{v_1, v_2\}$? If so, write it as a linear combination of $v_1$ and $v_2$. If not, say why not.


\bigskip

 \bf WEEK 6 \rm

 \bigskip

\item Determine whether the following sets of vectors are linearly independent or not.  Give a reason.  [Omitted]


\bigskip


\item Determine whether or not the following sets of vectors [in $\Re^3$]
form a basis.  [Omitted]

\bigskip



\end{enumerate}

\end{document}



\item The nullspace of the matrix
\[
A=\left[\begin{array}{cccc}
5 & -6 & 8 & -8   \\
-3 & 8 & 4 & -4  \\
2 & 2 & 12 & -12   
\end{array}\right] 
\]
has basis $B=\lc(-4,-2,1,0), (4,2,0,1) \rc$

\begin{enumerate}
\item Is $\ve{v}=(4,-2,1,1)$ in the nullspace of A? If so, find $\lb \ve{v}\rb_B$.
\item Is $\ve{w}=(20,10,-2,3)$ in the nullspace of A? If so, find $\lb \ve{w}\rb_B$.
\end{enumerate}


\item Let $B=\lc 2-x, 1-3x\rc$ be a (ordered) basis for $P_1$, and let $E=\lc 1, x\rc$ be the standard basis.
\begin{enumerate}
\item If $p=7-7x$, find $\lb p\rb_B$.
\item Find the change-of-basis matrix $P_{E\leftarrow B}$.
\item Find the change-of-basis matrix $P_{B\leftarrow E}$.
\end{enumerate}

\bigskip

\item Let $B=\lc (1,1,1), (-1,0,-1), (-1,-1,2) \rc$ be a (ordered) basis for $\Re^3$, and let $E=\lc (1,0,0),(0,1,0),(0,0,1)\rc$ be the standard basis.
\begin{enumerate}
\item Find the change-of-basis matrix $P_{E\leftarrow B}$.
\item Find the change-of-basis matrix $P_{B\leftarrow E}$.
\item If $\ve{v}=(2,4,-1)$, find $\lb \ve{v}\rb_B$.
\end{enumerate}


\bigskip


\item Consider the polynomials $p_1 = x^2+x-2$ and $p_2= -x^2+2x-1$ in $P_2$. Is the polynomial $3x^2-3$ in $\spn\{p_1, p_2\}$? If so, write it as a linear combination of $p_1$ and $p_2$. If not, say why not.

\item $\lc 1-x+x^2, 2-2x+x^2 \rc$ in $P_2$.   % lin indep

\item $\lc 1+x+x^2+x^3+x^4, 2x+2x^2-x^3-x^4, 2+x+x^2+2x^3-2x^4 \rc$ in $P_4$.
% basis

\bigskip

\end{enumerate}

\end{document}
\documentclass{article}

\setlength{\topmargin}{0pt}
\setlength{\oddsidemargin}{0pt}
\setlength{\evensidemargin}{0pt}
\setlength{\textwidth}{6in}
\setlength{\voffset}{0pt}
\setlength{\headheight}{0pt}
\setlength{\headsep}{0pt}
\setlength{\textheight}{9in}

\usepackage{amsmath}
\usepackage{latexsym}
\usepackage{amssymb}

\newcommand{\ve}{\vec}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\lb}{\left[}
\newcommand{\rb}{\right]}
\newcommand{\lc}{\left\{}
\newcommand{\rc}{\right\}}
\DeclareMathOperator{\curl}{curl}
\DeclareMathOperator{\diver}{div}
\newcommand{\ra}{\rightarrow}
\renewcommand{\Re}{\mathbb{R}}
\newcommand{\Nat}{\mathbb{N}}
\newcommand{\sB}{\mathcal{B}}
\newcommand{\sC}{\mathcal{C}}
\newcommand{\sF}{\mathcal{F}}
\renewcommand{\phi}{\varphi}
\DeclareMathOperator{\Real}{Re}
\DeclareMathOperator{\Imag}{Im}

\DeclareMathOperator{\spn}{span}


\begin{document}

\begin{center}
\LARGE
Homework \#6
\normalsize

\bigskip
Math 205

\bigskip
%{\bf Due Friday, Oct.\ 19 in class}
\end{center}
You must show your work in order to receive full credit; correct answers with no, or insufficient work, may not receive credit.
\bigskip

\normalsize

\begin{enumerate}
